Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9603158 · Physical Review Letters

Higher-Dimensional Bosonization with Nonlinear Energy Dispersion

Peter Kopietz, Guillermo E. Castilla

Published Jun 17, 1996• 31 citations
arXivHTMLPDF

Abstract

We develop a systematic method to treat the effect of nonlinearity in the energy dispersion on the usual bosonization result for the single-particle Green's function of fermions in arbitrary dimension. The leading corrections due to the quadratic term in the energy dispersion are explicitly calculated. In the Chern-Simons theory for half-filled quantum Hall systems, curvature is shown to be essential.

DOI
10.1103/physrevlett.76.4777

Related papers

Bosonization of interacting fermions in arbitrary dimension beyond the Gaussian approximation

Peter Kopietz, Joachim Hermisson, Kurt Schönhammer

We use our recently developed functional bosonization approach to bosonize interacting fermions in arbitrary dimension d beyond the Gaussian approximation. Even in d=1 the finite curvature of the energy dispersion at the Fermi surface gives rise to interactions between the bosons. In higher dimensions scattering processes describing momentum transfer between different patches on the Fermi surface (around-the-corner processes) are an additional source for corrections to the Gaussian approximation. We derive an explicit expression for the leading correction to the bosonized Hamiltonian and the irreducible self-energy of the bosonic propagator that takes the finite curvature as well as around-the-corner proceses into account. In the special case that around-the-corner scattering is negligible, we show that the self-energy correction to the Gaussian propagator is negligible if the dimensionless quantities (${\mathit{q}}_{\mathit{c}}$/${\mathit{k}}_{\mathit{F}}$${)}^{\mathit{d}}$${\mathit{F}}_{0}$ [1+${\mathit{F}}_{0}$${]}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\mu}/${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\Vert}\ensuremath{\partial}${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$/\ensuremath{\partial}\ensuremath{\mu}\ensuremath{\Vert} are small compared with unity for all patches \ensuremath{\alpha}. Here ${\mathit{q}}_{\mathit{c}}$ is the cutoff of the interaction in wave-vector space, ${\mathit{k}}_{\mathit{F}}$ is the Fermi wave vector, \ensuremath{\mu} is the chemical potential, ${\mathit{F}}_{0}$ is the usual dimensionless Landau interaction parameter, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$ is the local density of states associated with patch \ensuremath{\alpha}. We also show that the well-known cancellation between vertex and self-energy corrections in one-dimensional systems, which is responsible for the fact that the random-phase approximation (RPA) for the density-density correlation function is exact in d=1, exists also in d>1, provided (1) the interaction cutoff ${\mathit{q}}_{\mathit{c}}$ is small compared with ${\mathit{k}}_{\mathit{F}}$, and (2) the energy dispersion is locally linearized at the Fermi surface. Finally, we suggest a systematic method to calculate corrections to the RPA, which is based on the perturbative calculation of the irreducible bosonic self-energy arising from the non-Gaussian terms of the bosonized Hamiltonian.

arXiv:cond-mat/9502089•Physical review. B, Condensed matter•Oct 15, 1995•74 citations

HOW DOES A QUADRATIC TERM IN THE ENERGY DISPERSION MODIFY THE SINGLE-PARTICLE GREEN'S FUNCTION OF THE TOMONAGA–LUTTINGER MODEL?

Tom Busche, Peter Kopietz

We calculate the effect of a quadratic term in the energy dispersion on the low-energy behavior of the Green's function of the spinless Tomonaga–Luttinger model (TLM). Assuming that for small wave-vectors q= k-k F the fermionic excitation energy relative to the Fermi energy is v F q+q2/(2m), we explicitly calculate the single-particle Green's function for finite but small values of λ=q c /(2k F ). Here k F is the Fermi wave-vector, q c is the maximal momentum transfered by the interaction, and v F =k F /m is the Fermi velocity. Assuming equal forward scattering couplings g2=g4, we find that the dominant effect of the quadratic term in the energy dispersion is a renormalization of the anomalous dimension. In particular, at weak coupling the anomalous dimension is [Formula: see text], where γ is the anomalous dimension of the TLM. We also show how to treat the change of the chemical potential due to the interactions within the functional bosonization approach in arbitrary dimensions.

arXiv:cond-mat/9911479•International Journal of Modern Physics B•Jun 10, 2000•3 citations

Calculation of the single-particle Green's function of interacting fermions in arbitrary dimension via functional bosonization

Peter Kopietz

The single-particle Green's function of an interacting Fermi system with dominant forward scattering is calculated by decoupling the interaction by means of a Hubbard-Stratonowich transformation involving a bosonic auxiliary field $ϕ^α$. We obtain a higher dimensional generalization of the well-known one-dimensional bosonization result for the Green's function by first calculating the Green's function for a fixed configuration of the $ϕ^α$-field and then averaging the resulting expression with respect to the probability distribution ${\cal{P}} \{ ϕ^α \} \propto \exp [ - S_{eff} \{ ϕ^α \} ]$, where $S_{eff} \{ ϕ^α \}$ is the effective action of the $ϕ^α$-field. We emphasize the approximations inherent in the higher-dimensional bosonization approach and clarify its relation with diagrammatic perturbation theory.

arXiv:cond-mat/9506132•Jun 28, 1995•2 citations

Quasiparticle Behavior of Composite Fermions in the Half-Filled Landau Level

Peter Kopietz, Guillermo E. Castilla

We calculate the effect of infrared fluctuations of the Chern-Simons gauge field on the single-particle Green's function of composite fermions in the half-filled Landau level via higher-dimensional bosonization on a curved Fermi surface. We find that composite fermions remain well-defined quasiparticles, with an effective mass given by the mean-field value, but with anomalously large damping and a spectral function that contains considerable weight away from the quasiparticle peak.

arXiv:cond-mat/9611139•Physical Review Letters•Jan 13, 1997•18 citations

Theory of fermion liquids

H. Dharma Kwon, A. Houghton, John Bradley Marston

We develop a general theory of fermion liquids in spatial dimensions greater than 1. The principal method, bosonization, is applied to the cases of short- and long-range longitudinal interactions and to transverse gauge interactions. All the correlation funtions of the system may be obtained with the use of a generating functional. Short-range and Coulomb interactions do not destroy the Landau-Fermi fixed point. Non-Fermi liquid fixed points are found, however, in the cases of a super-long-range longitudinal interaction in two dimensions and transverse gauge interactions in two and three spatial dimensions. We consider in some detail the (2+1)-dimensional problem of a Chern-Simons gauge action combined with a longitudinal two-body interaction V(q)\ensuremath{\propto}\ensuremath{\Vert}q${\mathrm{\ensuremath{\Vert}}}^{\mathit{y}\mathrm{\ensuremath{-}}1}$, which controls the density, and hence gauge, fluctuations. For y0 we find that the gauge interaction is irrelevant and the Landau fixed point is stable, while for y>0 the interaction is relevant and the fixed point cannot be accessed by bosonization. Of special importance is the case y=0 (Coulomb interaction), which describes the Halperin-Lee-Read theory of the half-filled Landau level. We obtain the full quasiparticle propagator, which is of a marginal Fermi-liquid form. Using Ward identities, we show that neither the inclusion of nonlinear terms in the fermion dispersion nor vertex corrections alters our results: the fixed point is accessible by bosonization. As the two-point fermion Green's function is not gauge invariant, we also invetigate the gauge-invariant density response function. Near momentum Q=2${\mathit{k}}_{\mathit{F}}$, in addition to the Kohn anomaly we find other nonanalytic behavior. In the appendies we present a numerical calculation of the spectral function for a Fermi liquid with Landau parameter ${\mathit{f}}_{0}$\ensuremath{\ne}0. We also show how Kohn's theorem is satisfied within the bosonization framework.

arXiv:cond-mat/9501067•Physical review. B, Condensed matter•Sep 15, 1995•58 citations

Nonlinear Bosonization and Refermionization in One Dimension with the Keldysh Functional Integral

Filippo Bovo

We develop a self-contained approach to bosonization and refermionization using the Keldysh functional integral. Starting from fermionic particles, we bosonize the system and obtain a description in terms of the Tomonaga-Luttinger liquid, with, in addition, an infinite series of interaction terms arising from the curvature of the fermionic particle spectrum. We explicitly calculate the leading interaction term and check its consistency with a different approach based on the Matsubara framework, within which we calculate the second leading interaction term, as well. Moreover, we bosonize weakly and strongly interacting bosonic particles, and, finally, refermionize interacting phonons into non-interacting fermionic quasiparticles. The work culminates in a map between bosonic and fermionic particles and effective bosonic and fermionic excitations, representing phonons and fermionic quasiparticles.

arXiv:1610.08110•Oct 25, 2016•1 citation
We develop a systematic method to treat the effect of nonlinearity in the energy dispersion on the usual bosonization result for the single-particle Green's function of fermions in arbitrary dimension. The leading corrections due to the quadratic term in the energy dispersion are explicitly calculated. In the Chern-Simons theory for half-filled quantum Hall systems, curvature is shown to be essential.
We use our recently developed functional bosonization approach to bosonize interacting fermions in arbitrary dimension d beyond the Gaussian approximation. Even in d=1 the finite curvature of the energy dispersion at the Fermi surface gives rise to interactions between the bosons. In higher dimensions scattering processes describing momentum transfer between different patches on the Fermi surface (around-the-corner processes) are an additional source for corrections to the Gaussian approximation. We derive an explicit expression for the leading correction to the bosonized Hamiltonian and the irreducible self-energy of the bosonic propagator that takes the finite curvature as well as around-the-corner proceses into account. In the special case that around-the-corner scattering is negligible, we show that the self-energy correction to the Gaussian propagator is negligible if the dimensionless quantities (qc{\mathit{q}}_{\mathit{c}}qc​/kF{\mathit{k}}_{\mathit{F}}kF​)d{)}^{\mathit{d}})dF0{\mathit{F}}_{0}F0​ [1+F0{\mathit{F}}_{0}F0​${]}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\mu}/${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\Vert}\ensuremath{\partial}${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$/\ensuremath{\partial}\ensuremath{\mu}\ensuremath{\Vert} are small compared with unity for all patches \ensuremath{\alpha}. Here qc{\mathit{q}}_{\mathit{c}}qc​ is the cutoff of the interaction in wave-vector space, kF{\mathit{k}}_{\mathit{F}}kF​ is the Fermi wave vector, \ensuremath{\mu} is the chemical potential, F0{\mathit{F}}_{0}F0​ is the usual dimensionless Landau interaction parameter, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$ is the local density of states associated with patch \ensuremath{\alpha}. We also show that the well-known cancellation between vertex and self-energy corrections in one-dimensional systems, which is responsible for the fact that the random-phase approximation (RPA) for the density-density correlation function is exact in d=1, exists also in d>1, provided (1) the interaction cutoff qc{\mathit{q}}_{\mathit{c}}qc​ is small compared with kF{\mathit{k}}_{\mathit{F}}kF​, and (2) the energy dispersion is locally linearized at the Fermi surface. Finally, we suggest a systematic method to calculate corrections to the RPA, which is based on the perturbative calculation of the irreducible bosonic self-energy arising from the non-Gaussian terms of the bosonized Hamiltonian.
We calculate the effect of a quadratic term in the energy dispersion on the low-energy behavior of the Green's function of the spinless Tomonaga–Luttinger model (TLM). Assuming that for small wave-vectors q= k-k F the fermionic excitation energy relative to the Fermi energy is v F q+q2/(2m), we explicitly calculate the single-particle Green's function for finite but small values of λ=q c /(2k F ). Here k F is the Fermi wave-vector, q c is the maximal momentum transfered by the interaction, and v F =k F /m is the Fermi velocity. Assuming equal forward scattering couplings g2=g4, we find that the dominant effect of the quadratic term in the energy dispersion is a renormalization of the anomalous dimension. In particular, at weak coupling the anomalous dimension is [Formula: see text], where γ is the anomalous dimension of the TLM. We also show how to treat the change of the chemical potential due to the interactions within the functional bosonization approach in arbitrary dimensions.
The single-particle Green's function of an interacting Fermi system with dominant forward scattering is calculated by decoupling the interaction by means of a Hubbard-Stratonowich transformation involving a bosonic auxiliary field ϕαϕ^αϕα. We obtain a higher dimensional generalization of the well-known one-dimensional bosonization result for the Green's function by first calculating the Green's function for a fixed configuration of the ϕαϕ^αϕα-field and then averaging the resulting expression with respect to the probability distribution P{ϕα}∝exp⁡[−Seff{ϕα}]{\cal{P}} \{ ϕ^α \} \propto \exp [ - S_{eff} \{ ϕ^α \} ]P{ϕα}∝exp[−Seff​{ϕα}], where Seff{ϕα}S_{eff} \{ ϕ^α \}Seff​{ϕα} is the effective action of the ϕαϕ^αϕα-field. We emphasize the approximations inherent in the higher-dimensional bosonization approach and clarify its relation with diagrammatic perturbation theory.
We develop a general theory of fermion liquids in spatial dimensions greater than 1. The principal method, bosonization, is applied to the cases of short- and long-range longitudinal interactions and to transverse gauge interactions. All the correlation funtions of the system may be obtained with the use of a generating functional. Short-range and Coulomb interactions do not destroy the Landau-Fermi fixed point. Non-Fermi liquid fixed points are found, however, in the cases of a super-long-range longitudinal interaction in two dimensions and transverse gauge interactions in two and three spatial dimensions. We consider in some detail the (2+1)-dimensional problem of a Chern-Simons gauge action combined with a longitudinal two-body interaction V(q)\ensuremath{\propto}\ensuremath{\Vert}q${\mathrm{\ensuremath{\Vert}}}^{\mathit{y}\mathrm{\ensuremath{-}}1}$, which controls the density, and hence gauge, fluctuations. For y0 we find that the gauge interaction is irrelevant and the Landau fixed point is stable, while for y>0 the interaction is relevant and the fixed point cannot be accessed by bosonization. Of special importance is the case y=0 (Coulomb interaction), which describes the Halperin-Lee-Read theory of the half-filled Landau level. We obtain the full quasiparticle propagator, which is of a marginal Fermi-liquid form. Using Ward identities, we show that neither the inclusion of nonlinear terms in the fermion dispersion nor vertex corrections alters our results: the fixed point is accessible by bosonization. As the two-point fermion Green's function is not gauge invariant, we also invetigate the gauge-invariant density response function. Near momentum Q=2kF{\mathit{k}}_{\mathit{F}}kF​, in addition to the Kohn anomaly we find other nonanalytic behavior. In the appendies we present a numerical calculation of the spectral function for a Fermi liquid with Landau parameter f0{\mathit{f}}_{0}f0​\ensuremath{\ne}0. We also show how Kohn's theorem is satisfied within the bosonization framework.
We develop a self-contained approach to bosonization and refermionization using the Keldysh functional integral. Starting from fermionic particles, we bosonize the system and obtain a description in terms of the Tomonaga-Luttinger liquid, with, in addition, an infinite series of interaction terms arising from the curvature of the fermionic particle spectrum. We explicitly calculate the leading interaction term and check its consistency with a different approach based on the Matsubara framework, within which we calculate the second leading interaction term, as well. Moreover, we bosonize weakly and strongly interacting bosonic particles, and, finally, refermionize interacting phonons into non-interacting fermionic quasiparticles. The work culminates in a map between bosonic and fermionic particles and effective bosonic and fermionic excitations, representing phonons and fermionic quasiparticles.